Sample Quantiles Research Articles

ОЦЕНКА ПИКОВЫХ ЗНАЧЕНИЙ ПАРАМЕТРОВ КОЛЕБАТЕЛЬНЫХ ПРОЦЕССОВ

Estimation of machinery health, identification of defects and deviations of dynamic mechanical and technological equipment, formation and determination of parameters and criteria of technical condition are based on the measurement of parameters of oscillatory processes. As a rule, as a source of information vibroacoustic oscillatory processes are used. Such processes are characteristic both for vibroacoustic-based diagnostics, and for acoustic-emission control. Many state evaluation criteria are based on the measurement of process peak parameters. Given the random nature of the processes, estimating the measurement error and determining the peak values of signals becomes an urgent task. The task of the given work is presentation of methodical bases of estimation of measurement error and determination of peak values of vibroacoustic signals taking into account laws of distribution of their instantaneous values. In particular, the paper presents a solution to the problem of determining the dependence of the asymptotic error estimate of the sample quantile on the value of discrete values in the sample (time realization of the signal). This allows us to justify and verify the metrology of measuring instruments. Besides it allows to estimate reliability of measurements of peak values of the vibroacoustic signal.

A Risk Index and Data Display for Process Performance in the Pharmaceutical Industry.

We propose a new index and graphical display for quantifying and visualizing process performance in the pharmaceutical industry. These tools can provide management a comprehensive, high level overview of the process performance of a global manufacturing network suitable for risk ranking, by which is meant: identifying those processes at greatest risk of failing to meet specifications, and prioritizing resources to drive continuous process improvement. Our index, like others currently in use, compares the observed variation of CQAs-critical quality attributes-to their specifications. However, instead of relying on traditional data summaries such as means and standard deviations to characterize process results, the proposed index uses sample quantiles. Quantiles are more accurate and reliable when data are skewed or short-tailed as is often observed for pharmaceutical processes. Perhaps just as important, we communicate the results with a new visual display that accurately compares processes and sites. The display identifies instances when the summaries may mislead and the subject matter expert needs to "drill down" into manufacturing data to assure correct understanding.LAY ABSTRACT: The proposed risk index and graphical display enables high-risk processes to be identified, process improvements to be prioritized, resources to be efficiently allocated, and strategic planning for continuous process improvement to be evidence-based.

MODERATE DEVIATION PRINCIPLE OF SAMPLE QUANTILES AND ORDER STATISTICS

In this paper, we mainly study the moderate deviation principle of sample quantiles and order statistics for stationary m-dependent random variables. The results obtained in this paper extend the corresponding ones for an independent and identically distributed sequence to a stationary m-dependent sequence.

Large and moderate deviation principles for the bootstrap sample quantile

Let {X,Xn,n≥1} be a sequence of independent identically distributed (i.i.d.) random variables defined on a probability space (Ω,F,P) with common distribution F, and let {Xn,1∗,⋯,Xn,n∗} be a bootstrap sample from the empirical distribution Fn. Based on the bootstrap sample, the moderate deviation, large deviation and Bahadur’s asymptotic efficiency of the bootstrap sample pth quantile ξ̂np∗ are established.

A Unified Approach to Estimating and Testing Income Distributions With Grouped Data

We propose a unified approach that is flexibly applicable to various types of grouped data for estimating and testing parametric income distributions. To simplify the use of our approach, we also provide a parametric bootstrap method and show its asymptotic validity. We also compare this approach with existing methods for grouped income data, and assess their finite-sample performance by a Monte Carlo simulation. For empirical demonstrations, we apply our approach to recovering China's income/consumption distributions from a sequence of income/consumption share tables and the U.S. income distributions from a combination of income shares and sample quantiles. Supplementary materials for this article are available online.

K-quantiles: L1 distance clustering under a sum constraint

This paper considers centroid-based clustering under ℓ1 distance in which both data points and cluster centers are subject to a sum constraint on their components. A closed-form solution is derived for the cluster center optimization problem, enabling an interpretation as a sample quantile of the cluster. An adaptive sampling initialization step is also adopted to provide a guarantee on expected clustering cost as well as empirical improvements. Experiments on synthetic data indicate that the advantages of the proposed algorithms increase as clusters become more concentrated and as the dimension increases. An application to clustering employee job role profiles highlights the utility of ℓ1 distance in promoting sparse, interpretable cluster centers.

Risk Measure Estimates in Quiet and Turbulent Times: An Empirical Study

In this study we empirically explore the capacity of historical VaR to correctly predict the future risk of a financial institution. We observe that rolling samples are better able to capture the dynamics of future risks. We thus introduce another risk measure, the Sample Quantile Process, which is a generalization of the VaR calculated on a rolling sample, and study its behavior as a predictor by varying its parameters. Moreover, we study the behavior of the future risk as a function of past volatility. We show that if the past volatility is low, the historical computation of the risk measure underestimates the future risk, while in period of high volatility, the risk measure overestimates the risk, confirming that the current way financial institutions measure their risk is highly procyclical.

An ergodic theorem for proportions of observations that fall into random sets determined by sample quantiles

Assume that a sequence of observations (X_n; nge 1) forms a strictly stationary process with an arbitrary univariate cumulative distribution function. We investigate almost sure asymptotic behavior of proportions of observations in the sample that fall into a random region determined by a given Borel set and a sample quantile. We provide sufficient conditions under which these proportions converge almost surly and describe the law of the limiting random variable.

A Bootstrap Test for Symmetry based on Quantiles

In this paper, we construct a nonparametric test for the symmetry assumption of an underling distribution based on the sample quantiles. Bootstrap re-sampling from a symmetric empirical distribution function is used to obtain the p-value of the test. The power of the new test statistic is compared with some well-known symmetry tests using a simulation study. The results show that the proposed test preserves its level and it has reasonable power properties on the family of distribution evaluated.

Growth of tuber crops and almost sure band for quantiles

ABSTRACTSome tuber crops are governed by memoryless property of exponential distribution leading to a mixture distribution with heavy tail. Quantile-based estimators may then be appropriate than mean as a measure of central tendency. We prove almost sure representation theorems for sample quantiles in a general setup of U statistics, under slightly stronger assumption than assuming the existence of a continuously differentiable distribution function F for the kernel h. We obtain almost sure (a.s.) upper and lower estimate for F− 1(p), p ∈ (0, 1) as a band for p varying. As an application, dataset arising from two varieties of potato cultivation are analyzed.

Berry–Esséen theorem for sample quantiles of asymptotically uncorrelated non reversible Markov chains

ABSTRACTWe prove Berry–Esséen bound for sample quantiles of Markov chains with spectral gap in L2.

Easy and accurate variance estimation of the nonparametric estimator of the partial area under the ROC curve and its application.

The receiver operating characteristic (ROC) curve is a popular technique with applications, for example, investigating an accuracy of a biomarker to delineate between disease and non-disease groups. A common measure of accuracy of a given diagnostic marker is the area under the ROC curve (AUC). In contrast with the AUC, the partial area under the ROC curve (pAUC) looks into the area with certain specificities (i.e., true negative rate) only, and it can be often clinically more relevant than examining the entire ROC curve. The pAUC is commonly estimated based on a U-statistic with the plug-in sample quantile, making the estimator a non-traditional U-statistic. In this article, we propose an accurate and easy method to obtain the variance of the nonparametric pAUC estimator. The proposed method is easy to implement for both one biomarker test and the comparison of two correlated biomarkers because it simply adapts the existing variance estimator of U-statistics. In this article, we show accuracy and other advantages of the proposed variance estimation method by broadly comparing it with previously existing methods. Further, we develop an empirical likelihood inference method based on the proposed variance estimator through a simple implementation. In an application, we demonstrate that, depending on the inferences by either the AUC or pAUC, we can make a different decision on a prognostic ability of a same set of biomarkers. Copyright © 2016 John Wiley & Sons, Ltd.

Regressing sample quantiles to perform nonparametric capability analysis

Capability analysis corresponds to a set of methods used to estimate and test the ability of an in-control process to provide a specific output. When there is only one quality characteristic that behaves as a continuous random variable, indices like C p and C pk can be used to measure how well requirements are met. Under normality, variation is indicated using 3−s i g m a limits; otherwise, the corresponding quantiles are used. Distribution fitting and transformations to normality can be used to estimate quantiles by finding an overall fit to the data available. However, by giving the same weight to all observations, the best possible fit of extreme values can be lost. To address this issue, a regression approach is proposed to fit functions over maximum likelihood estimates of probabilities of extreme values. A case study from the automotive industry is used to illustrate the proposed approach. To evaluate the performance, extensive Monte Carlo simulation is used, and the results are compared with the corresponding approach using the Clements method. The proposed nonparametric technique shows smaller MAD when high levels of skewness exist. Practitioners with basic knowledge of regression analysis may find the approach useful to estimate capability indices without the need of a specific probability distribution.

A modified Q-Q plot for large sample sizes

The Q-Q plot is a graphical tool for assessing the fit of observed data to a theoretical distribution, in which every single observation in the data is represented by a symbol (usually a dot). In many occasions, due either to natural variations of the data or to a large sample size, the Q-Q plot could be interpreted as a sign of failure of the proposed model. An alternative is to consider a special set of characteristics of the data such as the sample quantiles that, jointly with its theoretical counterparts, allow the user to effectively compare both. We propose and illustrate a modified Q-Q plot that helps to visualise the differences between the observed quantiles and its corresponding theoretical values, and overcome some technical problems of the traditional Q-Q plot.

Assessing a hypothesis test for the difference between two quantiles from independent populations

ABSTRACTAn empirical distribution function estimator for the difference of order statistics from two independent populations can be used for inference between quantiles from these populations. The inferential properties of the approach are evaluated in a simulation study where different sample sizes, theoretical distributions, and quantiles are studied. Small to moderate sample sizes, tail quantiles, and quantiles which do not coincide with the expectation of an order statistic are identified as problematic for appropriate Type I error control.

The Bahadur representation of sample quantiles for weakly dependent sequences

Under some weak conditions, we use the moment inequality for -mixing sequence to investigate the Bahadur representation of sample quantiles based on this stochastic sequence. For all , we obtain a rate of Bahadur representation in the form of , which is close to the rate .

Bayesian Curve-Fitting in Semiparametric Small Area Models with Measurement Errors

We study a semiparametric Bayesian approach to small area estimation under a nested error linear regression model with area level covariate subject to measurement error. Consideration is given to radial basis functions for the regression spline and knots on a grid of equally spaced sample quantiles of covariate with measurement errors in the nested error linear regression model setup. We conduct a hierarchical Bayesian structural measurement error model for small areas and prove the propriety of the joint posterior based on a given hierarchical Bayesian framework since some priors are defined non-informative improper priors that uses Markov Chain Monte Carlo methods to fit it. Our methodology is illustrated using numerical examples to compare possible models based on model adequacy criteria; in addition, analysis is conducted based on real data.

Geometric ergodicity of random scan Gibbs samplers for hierarchical one-way random effects models

We consider two Bayesian hierarchical one-way random effects models and establish geometric ergodicity of the corresponding random scan Gibbs samplers. Geometric ergodicity, along with a moment condition, guarantees a central limit theorem for sample means and quantiles. In addition, it ensures the consistency of various methods for estimating the variance in the asymptotic normal distribution. Thus our results make available the tools for practitioners to be as confident in inferences based on the observations from the random scan Gibbs sampler as they would be with inferences based on random samples from the posterior.

Bayesian curve-fitting with radial basis functions under functional measurement error model

Abstract This article presents Bayesian approach to regression splines with knots on a grid ofequally spaced sample quantiles of the independent variables under functional measure-ment error model. We consider small area model by using penalized splines of non-linearpattern. Speci cally, in a basis functions of the regression spline, we use radial basisfunctions. To t the model and estimate parameters we suggest a hierarchical Bayesianframework using Markov Chain Monte Carlo methodology. Furthermore, we illustratethe method in an application data. We check the convergence by a potential scale re-duction factor and we use the posterior predictive p value and the mean logarithmicconditional predictive ordinate to compar models.Keywords: Functional, hierarchical Bayes, measurement error, radial basis, semipara-metric. 1. Introduction We developed the semiparametric small area models with measurement errors models inour previous paper (Hwang and Kim, 2010). Speci cally, we considered small area model byusing penalized splines of non-linear pattern based on truncated polynomial basis functionsand knots on a grid of equally spaced sample quantiles under functional measurement errormodel. Measurement error modeling is also related to Goo and Kim (2013).The truncated polynomial basis functions (TPBF) is simple, but not always numericallystable when the number of knots is large and the smoothing parameter close to zero. In thiscase the computation has to be organized carefully and numerically superior alternativesare available, like B-splines and radial basis functions (Ruppert et al., 2003).The objective of this article is to develop alternative estimators of small area means byusing radial basis functions (RBF) with functional measurement error model. RBF is de nedwith degree pfrom Ruppert et al. (2003) as follows.1;x; ;x

A NEW CHARACTERIZATION OF THE NORMAL DISTRIBUTION AND TEST FOR NORMALITY

We study the asymptotic covariance function of the sample mean and quantile, and derive a new and surprising characterization of the normal distribution: the asymptotic covariance between the sample mean and quantile is constant across all quantiles,if and only ifthe underlying distribution is normal. This is a powerful result and facilitates statistical inference. Utilizing this result, we develop a new omnibus test for normality based on the quantile-mean covariance process. Compared to existing normality tests, the proposed testing procedure has several important attractive features. Monte Carlo evidence shows that the proposed test possesses good finite sample properties. In addition to the formal test, we suggest a graphical procedure that is easy to implement and visualize in practice. Finally, we illustrate the use of the suggested techniques with an application to stock return datasets.